# How to solve the nonlinear higher-order PDEs with overdetermined quantities of conditions whose their most-general solutions are known?

Some nonlinear higher-order PDEs have nice form of most-general solutions, for example the nonlinear second-order PDE $u_{tt}=a^2u_{xx}+be^{cu}$ where $a,b,c\neq0$ , according to http://eqworld.ipmnet.ru/en/solutions/npde/npde2103.pdf, it has the nice form of most-general solution $u(x,t)=\dfrac{f(x-at)+g(x+at)}{c}-\dfrac{2}{c}\ln\left(k\int^{x-at}e^{f(r)}~dr-\dfrac{bc}{8a^2k}\int^{x+at}e^{g(s)}~ds\right)~.$

If there are the conditions of which the quantities not more than the quantities of the arbitrary functions, we can easily solve for the arbitrary functions. Now, how about the issues of the conditions of which the quantities more than the quantities of the arbitrary functions, i.e. for examples how to solve the above PDE with the conditions respectively?

(i) $u(0,t)=0$ , $u_x(0,t)=0$ , $u(x,0)=p(x)$ , $u_t(x,0)=q(x)$

(ii) $u(0,t)=0$ , $u(\alpha,t)=0$ , $u(x,0)=p(x)$ , $u_t(x,0)=q(x)$

(iii) $u(0,t)=0$ , $u(\alpha,t)=0$ , $u(x,0)=p(x)$ , $u(x,\beta)=q(x)$

I can handle these issues when the PDE is linear, but I get stuck on the issues when the PDE is nonlinear.

I know that in these cases piecewise solution should be introduced, but how should I introduce the piecewise solution suitably in these cases? Can I partly follow the approach in Wave Equation with One Non-Homogeneous Boundary Condition?

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